On the spectral properties of a non-coercive mixed problem associated with $\overline\partial$-operator

We consider a non-coercive Sturm–Liouville boundary value problem in a bounded domain $D$ of the complex space $\mathbb C^n$ for the perturbed Laplace operator. More precisely, the boundary conditions are of Robin type on $\partial D$ while the first order term of the boundary operator is the complex normal derivative. We prove that the problem is Fredholm one in proper spaces for which an Embedding Theorem is obtained; the theorem gives a correlation with the Sobolev–Slobodetskii spaces. Then, applying the method of weak perturbations of compact self-adjoint operators, we show the completeness of the root functions related to the boundary value problem in the Lebesgue space. For the ball, we present the corresponding eigenvectors as the product of the Bessel functions and the spherical harmonics.